A naive Bayes classifier picks the class with the highest posterior probability given a point's features, using Bayes' rule and one simplifying assumption.

Bayes' rule

P(c | x) ∝ P(c) · P(x | c). The class prior P(c) is the fraction of training points in class c; the likelihood P(x | c) says how well the point fits that class.

The "naive" assumption

Features are assumed conditionally independent given the class, so the joint likelihood factorises: P(x | c) = P(x₁ | c) · P(x₂ | c). It is rarely exactly true, yet the classifier is fast and works surprisingly well.

Gaussian model

For continuous features we fit a Gaussian per class per axis: P(xⱼ | c) = 𝒩(xⱼ; μ_{cj}, σ²_{cj}). Each class therefore has a mean and variance on each axis, estimated from its points. The shaded ellipse marks one standard deviation.

Decision boundary

The background colours every pixel by the argmax posterior class; the more confident the model, the stronger the shade. Because variances differ between classes the boundary is quadratic, not a straight line — that is the signature of Gaussian naive Bayes.